Question

Which of the following statements are correct? Choose all that apply.a. lim x→1 1/ (x−$$1)^2$$ does not existb. lim x→1 1/ (x−$$1)^2$$=∞c. lim x→1 1/(x−$$1)^2$$=−∞

Accepted Answer

Step 1: Understand the expression given in the limit: $$ \lim_{x 	o 1} \frac{1}{(x-1)^2} . $$This is a rational function where the denominator approaches zero as $$ x $$ approaches 1. Step 2: Analyze the behavior of the function as $$ x $$ approaches 1 from the left ($$ x 	o 1^- ) $$and from the right ($$ x 	o 1^+ ). In $$both cases, $$ (x-1)^2 $$ approaches zero, but since it is squared, it remains positive. Step 3: Since the denominator $$ (x-1)^2 $$ approaches zero and is always positive, the fraction $$ \frac{1}{(x-1)^2} $$ will increase without bound, indicating that the limit approaches infinity. Step 4: Evaluate the correctness of each statement: (a) The limit does not exist if it approaches different values from the left and right, but here it approaches the same value (infinity) from both sides. (b) The limit approaches infinity, which is consistent with our analysis. (c) The limit cannot be negative infinity because the denominator is always positive. Step 5: Conclude that statement (b) is correct, as the limit approaches infinity, while statement (a) is incorrect because the limit does exist (it is infinity), and statement (c) is incorrect because the limit is not negative infinity.

Which of the following statements are correct? Choose all that apply.

a. lim x→1 1/ (x−1)^2 does not exist

b. lim x→1 1/ (x−1)^2=∞

c. lim x→1 1/(x−1)^2=−∞

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Key Concepts

Limits

Infinite Limits

Behavior of Rational Functions

Which of the following statements are correct? Choose all that apply.a. lim x→1 1/ (x−1)^2 does not existb. lim x→1 1/ (x−1)^2=∞c. lim x→1 1/(x−1)^2=−∞